Electron. J. Diff. Equ., Vol. 2010(2010), No. 131, pp. 1-11.

Oscillation of solutions for third order functional dynamic equations

Elmetwally M. Elabbasy, Taher S. Hassan

Abstract:
In this article we study the oscillation of solutions to the third order nonlinear functional dynamic equation
$$ L_{3}(x(t))+\sum_{i=0}^{n}p_i(t)\Psi_k{\alpha_ki}(x(h_i(t)))=0, $$
on an arbitrary time scale $\mathbb{T}$. Here
$$ L_0(x(t))=x(t),\quad L_k(x(t))=\Big(\frac{[ L_{k-1}x(t)]^{\Delta }}{a_k(t)}\Big)^{\gamma_kk}, \quad k=1,2,3 $$
with $a_1, a_2$ positive rd-continuous functions on $\mathbb{T}$ and $a_{3}\equiv 1$; the functions $p_i$ are nonnegative rd-continuous on $\mathbb{T}$ and not all $p_i(t)$ vanish in a neighborhood of infinity; $\Psi_k{c}(u)=|u|^{c-1}u$, $c>0$. Our main results extend known results and are illustrated by examples.

Submitted April 13, 2010. Published September 14, 2010.
Math Subject Classifications: 34K11, 39A10, 39A99.
Key Words: Oscillation; third order; functional dynamic equations; time scales.

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Elmetwally M. Elabbasy
Department of Mathematics, Faculty of Science
Mansoura University, Mansoura, 35516, Egypt
email: emelabbasy@mans.edu.eg
http://www.mans.edu.eg/pcvs/10805/
Taher S. Hassan
Department of Mathematics, Faculty of Science
Mansoura University, Mansoura, 35516, Egypt
email: tshassan@mans.edu.eg
http://www.mans.edu.eg/pcvs/10805/

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